This is an exercise in Riemannian geometry. In this note I Taylor-expand the Riemannian metric in a normal neighborhood around a point in a Riemannian manifold . What I have done is nothing except providing a few more terms than most standard textbooks. Hopefully the computation is correct. (I haven’t checked against other sources. )

Theorem 1In a normal coordinates neighborhood of , the Taylor series of around is given by

*Proof:* In normal coordinates, fix for the moment and let be a radial geodesic, then is a Jacobi field on , where . (This can be seen by observing that is the variational vector field of the -family of geodesics , with a slight abuse of notations. ) Let .

We will use to denote a covariant derivative w.r.t. and define , then implies is a symmetric operator. Differentiating implies are also symmetric. Also, the Jacobi equation simply becomes

We compute (a way to test the correctness of the following computation is to note that each carries two derivatives)

(Actually to compute we don’t have to compute all the terms of : we can omit the term containing any ). It follows that

All the above expression are evaluated at , noting that . So we have (all repeated indices are summed over):

We thus conclude that

Here is the radial distance from .

The lengths of the formulae are remarkable. Thanks for KKK’s hard work in typesetting them.

in other places (for example http://arxiv.org/abs/1206.1092) the O(x^2) term has the opposite sign to the result here (j and l exchanged)

While I haven’t checked the higher order terms, the terms up to the forth order are correct, see e.g. Hamilton’s Ricci Flow p.59. The reason for the difference is that I have used a different convention for the Riemannian curvature tensor (my apology for not stating it): my definition is .

Thanks!

which can answer to my question: how to calculate the inverse of the metric g_ {ij}, in other words how to express g ^ {ij} -Thank you